Normal Distribution Calculator – Gaussian Distribution
Calculate probabilities and z-scores for normal distribution
How to Use
- Enter the mean (μ) of your distribution
- Enter the standard deviation (σ)
- Enter the X value you want to evaluate
- View the z-score and probabilities
What is Normal Distribution?
The normal distribution, also known as the Gaussian distribution or bell curve, is a probability distribution that is symmetric about the mean. It describes how values of a variable are distributed and is one of the most important probability distributions in statistics.
The normal distribution is characterized by two parameters: the mean (μ) which determines the center of the distribution, and the standard deviation (σ) which determines the spread or width of the distribution.
Understanding Z-Score
The z-score (or standard score) indicates how many standard deviations an element is from the mean. It is calculated as: z = (X - μ) / σ
- A z-score of 0 indicates the value is exactly at the mean
- A positive z-score indicates the value is above the mean
- A negative z-score indicates the value is below the mean
- About 68% of values fall within 1 standard deviation (z = ±1)
- About 95% of values fall within 2 standard deviations (z = ±2)
- About 99.7% of values fall within 3 standard deviations (z = ±3)
Applications of Normal Distribution
Normal distribution is widely used in:
- Quality control and Six Sigma methodologies
- Standardized testing and educational assessment
- Financial modeling and risk analysis
- Natural and social sciences research
- Machine learning and data science
- Hypothesis testing and confidence intervals
The 68-95-99.7 Rule
Also known as the empirical rule, this describes how data is distributed in a normal distribution:
- 68% of data falls within 1 standard deviation of the mean
- 95% of data falls within 2 standard deviations of the mean
- 99.7% of data falls within 3 standard deviations of the mean
Frequently Asked Questions
- What is the difference between normal distribution and standard normal distribution?
- A standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to a standard normal distribution using z-scores.
- How do I interpret the z-score?
- The z-score tells you how many standard deviations away from the mean your value is. For example, a z-score of 2 means the value is 2 standard deviations above the mean, which is relatively rare (only about 2.5% of values are higher).
- What does P(X < x) mean?
- P(X < x) represents the cumulative probability - the probability that a randomly selected value from the distribution will be less than x. This is also called the cumulative distribution function (CDF).
- When can I assume data follows a normal distribution?
- Many natural phenomena approximate normal distribution, especially when dealing with averages or sums of many independent random variables (Central Limit Theorem). However, always verify with statistical tests or visual inspection before assuming normality.